Let $M$ be an oriented smooth $n$-manifold, let $I \subset \mathbb{R}$ be an open interval, and let $g: I \to \Gamma(S^2T^*M)$ be a smooth one-parameter family of Riemannian metrics. For each $t \in I$, define the symmetric $2$-tensor $v(t) \in \Gamma(S^2T^*M)$ by $v(t) := \partial_t g(t)$. If $d\mu_{g(t)}$ denotes the Riemannian volume form of $g(t)$, then

\begin{align*} \partial_t d\mu_{g(t)} = \frac{1}{2}\operatorname{tr}_{g(t)}(v(t))\, d\mu_{g(t)}. \end{align*}

In particular, if $g(t)$ evolves by Ricci flow,

\begin{align*} \partial_t g(t) = -2\operatorname{Ric}_{g(t)}, \end{align*}

then

\begin{align*} \partial_t d\mu_{g(t)} = -S_{g(t)}\, d\mu_{g(t)}, \end{align*}

where $S_{g(t)} := \operatorname{tr}_{g(t)}\operatorname{Ric}_{g(t)}$ is the scalar curvature of $g(t)$.